51=18y^2+9y

Solution for 51=18y^2+9y equation:

51=18y^2+9y

We move all terms to the left:

51-(18y^2+9y)=0

We get rid of parentheses

-18y^2-9y+51=0

a = -18; b = -9; c = +51;
Δ = b2-4ac
Δ = -92-4·(-18)·51
Δ = 3753
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$y_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$y_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{3753}=\sqrt{9*417}=\sqrt{9}*\sqrt{417}=3\sqrt{417}$
$y_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-9)-3\sqrt{417}}{2*-18}=\frac{9-3\sqrt{417}}{-36}
$
$y_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-9)+3\sqrt{417}}{2*-18}=\frac{9+3\sqrt{417}}{-36}
$